Georgia Institute of Technology College of Sciences

Proving conjectures is an essential part of our job as mathematicians. Another essential part is to come up with plausible conjectures. In the talk, we focus on this part. We present a new twist to an old method from computer algebra for detecting recurrence equations of infinite sequences of which only the first few terms are known. By applying this new version systematically to all the entries of the Online Encyclopedia of Integer Sequences, we detected a number of potential recurrence equations that could not be found by the classical methods. Some of these have meanwhile been proven. This is joint work with Christoph Koutschan. 

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Titile: Lattice Reduction 
                                                                                                           
Abstract: It is well known how to go from an exact number (e.g. 1/3) into an approximation (e.g. 0.333). But how can we get back? At first glance, this seems impossible, because some information got lost during the approximation. However, there are techniques for doing this and similar seemingly magic tricks. We will discuss some such tricks that rely on an algorithm for finding short vectors in integer lattices.          

MacPhersonians are a combinatorial analog of real Grassmannians defined by oriented matroids.  A long standing conjecture says that each MacPhersonian is homotopy equivalent to the corresponding Grassmannian.  Pseudolinear Grassmannians are spaces of topological representations of oriented matroids, and these are each homotopy equivalent to the corresponding Grassmannian in rank 3.  I will present a good cover of the rank 3 pseudolinear Grassmannian with nerve complex isomorphic to the order complex of the corresponding MacPhersonian, confirming the conjecture in rank 3.

Power-flow equations model the intricate balancing conditions in electric power grids and are central to analysis and control.  They can be reformulated as Laurent polynomial systems, which makes algebraic and polyhedral techniques applicable.  In this talk, we first explore different ways in which this can be done.

However, certain algebraic formulations may be deficient: the actual number of isolated solutions (counting multiplicity) may fall below the Bernshtein–Kushnirenko–Khovanskii (BKK) bound predicted from Newton polytopes.  By choosing a proper parametrization one uncovers that this deficiency exhibits a certain toric structure.  Recognizing that structure reframes the deficit as a geometric feature rather than a numerical anomaly.  In the second part of this talk, we explore variations of polyhedral homotopy methods designed to respect and exploit this structure.

====(Below is the information on the pre-talk.)====

Title: Mixed volume, mixed cells, and stable self intersections

Abstract: This talk provides an introduction to mixed volume, mixed cells, and their connections to the Bernshtein–Kushnirenko–Khovanskii bound, as well as stable intersections of tropical hypersurfaces.

There are two standard ways to associate a principally polarized abelian variety (ppav) to a smooth algebraic curve X of genus g. The Jacobian variety Jac(X) is a ppav of dimension g. An etale double cover X’->X determines the Prym variety Prym(X’/X), which is a ppav of dimension g-1. These two objects are related by Recillas’ trigonal construction: given an etale double cover X’->X of a trigonal curve X, we can construct a tetragonal curve Y such that Prym(X’/X) is isomorphic to Jac(Y).

I will talk about a tropical version of the trigonal construction, where algebraic curves are replaced by metric graphs and ppavs by real tori with integral structure. Given a double cover X’->X of a trigonal graph X, we obtain a tetragonal graph Y such that the tropical Prym variety Prym(X’/X) and the tropical Jacobian Jac(Y) are isomorphic.

This construction has two applications. First, we can use it to compute the second moment of the tropical Prym variety for g up to 4, and conjecturally for all g, which has arithmetic applications. Second, the tropical trigonal construction provides an explicit resolution of the Prym—Torelli map in genus 4.